CHAPTER FIVE Surface Runoff

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    CHAPTER FIVE Surface Runoff


    Rainfall excess is that portion of total rainfall that is not stored on the land surface or infiltrated into underlying soil. It eventually comprises direct runoff to downstream rivers, streams, storm sewers, and other conveyance systems. One of the key parameters in the design and analysis of urban hydrologic systems is the resulting peak runoff or, in some cases, the variation of runoff over time (i.e., hydrograph) at a watershed outlet or other downstream design point. Its evaluation requires an adequate understanding of the processes and routes by which the transformation of excess rainfall to direct runoff occurs.

    It is worth noting that the models described herein for runoff estimation are characterized as lumped methods. In this respect, they use a single set of characteristic parameters to describe an entire basin. For example, the unit hydrograph methods assume rainfall excess to be uniform across a watershed, capable of being described by a single hyetograph. Strictly speaking, such parameters vary spatially; however, it is generally not feasible to account for the variability (e.g., runoff from every lawn, street, or roof), as in the case of a distributed model.


    Various parameters are used to characterize the response of a watershed to a rainfall event. Of these, the most frequently used is time of concentration, tc, defined as the time required for water to travel from the most hydraulically-remote portion of a watershed to a location of interest (e.g., basin outlet). This also corresponds to the response time from the beginning of a storm event to a time when the entire basin contributes runoff to that location. Beyond the time of concentration, runoff will remain constant until rainfall excess ceases.

    Because tc cannot be directly measured, it is often estimated based on travel times along appropriately partitioned flow paths that include both overland flow and more concentrated channel flow components. Flow times can depend on physiographic factors such as watershed size, topography, and land use. Climatic factors, such as rainfall intensity and duration, play an equally important role. For urban drainage systems, the timing of runoff may also be heavily influenced by a storm water collection system. In this case, time of concentration should be adjusted to reflect the additional travel time through the conveyance system.


    Numerous empirical and physically-based methods are available for approximating time of concentration. A study conducted by McCuen et al. (1984), however, compared these various methods and showed that results can vary significantly. Thus, the choice of which method to apply is not an easy one for the analyst. The selection requires careful consideration of limitations of each method and the conditions under which it was derived. In addition, it is often recommended to estimate tc using several methods and use qualitative judgment to arrive at a single value. Note that in practice, a minimum value of five minutes is recommended for urban applications and ten minutes for residential applications.

    5.2.1 Overland Flow Models

    Overland flow is that portion of runoff that occurs as sheet flow over a land surface without becoming concentrated in well-defined channels, gullies, and rills. A common example is flow over long, gradually-sloped pavements during or immediately following a storm. Some of the methods used for computing tc for overland flow include the kinematic wave model, NRCS models, and a variety of empirical techniques. Kinematic Wave Model

    Kinematic wave theory relies on the continuity equation (i.e., conservation of mass) and a simplified form of the momentum equation to derive solutions for flow problems. For a unit width of overland flow, the former can be expressed as

    (5-1) where q is the unit width flow rate; x is longitudinal distance along the flow path; y is flow depth; t is time; and i is the rate of rainfall, or rainfall excess in the case that abstractions are considered. Note that the two terms on the left-hand side of Equation 5-1 are used to simulate the non-uniform and unsteady flow aspects (i.e., spatial and temporal variation of flow), respectively. With respect to momentum, however, flow is assumed to be steady and uniform from one time increment to the next. As described in greater detail in Chapter 6, the implication is that simulated kinematic waves will not appreciably accelerate and can only flow in the downstream direction. Thus, a wave will be observed as relatively uniform rise and fall in the water surface over a long period. The method is, therefore, limited to conditions that do not demonstrate appreciable attenuation.


    xq =



    Consider that, for uniform flow, the momentum equation can be expressed in the general form

    (5-2) where k and m are constants that depend on a relationship between depth and discharge, Q. For example, the Manning equation represents one such relationship and can be expressed as

    (5-3) where Km is a constant equal to 1.49 in U.S. customary units and 1.0 in S.I. units; n is the Manning roughness coefficient; A is effective flow area; R is the hydraulic radius, defined as the ratio of flow area to wetted perimeter, P; and S is surface slope in ft/ft or m/m. Since overland flow can be considered as shallow flow through a very wide rectangular channel, hydraulic radius can be approximated as depth, and Equation 5-3 can be rewritten as

    (5-4) where B is flow width. In addition, to be consistent with the current kinematic wave analysis, Equation 5-4 can be expressed for a unit width as

    (5-5) Comparing Equations 5-5 and 5-2, k can be taken as

    (5-6) and m is 5/3. Combining these two expressions yields the kinematic wave equation for overland flow

    mk yq =

    2132m SARn

    KQ =

    2135m Syn

    Kq =

    ( ) 2132m SyByn

    KQ =

    21mk Sn





    c SinL938.0t =



    c iLt



    where ck is the kinematic wave celerity (i.e., speed), equal to

    (5-8) Equation 5-7 implies that to an observer moving at a speed (i.e., dx/dt) equivalent to ck, the relationship between depth of flow and rainfall excess is

    (5-9) Solving this relationship at initial conditions y = 0 everywhere at t = 0 yields

    (5-10) Substituting this result into Equation 5-8 and integrating subject to the boundary condition x = 0 at t = 0 gives

    (5-11) This expression can be used to evaluate the time required for a kinematic wave to travel an overland flow path of distance L, which is assumed to be equal to the time of concentration. The corresponding general relationship is


    Using the Manning equation to relate depth and discharge (i.e., m = 5/3 and k by Equation 5-6) (Morgali and Linsley, 1965; Aron and Erborge, 1973),



    xyck =


    1mkk myc


    idtdy =

    m1mk tix


    ity =




    c SPnL42.0t =

    where tc is in minutes; L is ft; n can be read from Table 5-1; i is in in/hr and is assumed to be uniform over the catchment; and S is in ft/ft. Note that because the slope is considered to be constant over L, the time of concentration should be computed and summed over relatively small topographic contour intervals.

    Table 5-1: Manning roughness coefficients for overland flow surfaces

    Surface description Manning n

    Concrete, asphalt 0.010 0.016

    Bare sand 0.010 0.016

    Gravel 0.012 0.030

    Bare clay-loam (eroded) 0.012 0.033

    Natural rangeland 0.010 0.320

    Bluegrass sod 0.39 0.63

    Short-grass prairie 0.10 0.20

    Dense grass, Bermuda grass, bluegrass 0.17 0.48

    Forestland 0.20 0.80 Since the time of concentration and rainfall intensity are both unknown,

    application of Equation 5-13 is iterative. An initial intensity is assumed and the corresponding value of tc is computed. The assumed value of i must be checked by determining a new time of concentration based on intensity-duration-frequency (IDF) relationships and comparing the new value with that previously computed. This process is repeated until values for intensity in successive iterations converge. Overton and Meadows (1976) used a power law to relate rainfall intensity and duration in order to bypass the need for this iterative solution. By substituting the 2-year, 24-hour rainfall depth, P24, for i, they proposed


    where P24 is in inches and can be obtained from corresponding IDF data. Since the kinematic wave model represented by Equations 5-13 and 5-14

    was derived using Manning equation, it is inherently limited to turbulent flows. Furthermore, the method assumes that no local inflow occurs; no backwater or storage effects are present; the discharge varies only with depth; and that


    ( )[ ]21


    c S1909CN1000Lt =

    V60Ltc =






    planar, non-converging flow predominates. Unfortunately, these assumptions become less realistic as land slope decreases, surface roughness increases, or the length of flow path increases (McCuen, 1998). Practical upper limits on length range from 100 to 300 ft (30 to 90 m). NRCS Methods

    The Natural Resources Conservation Service (NRCS), formerly the Soil Conservation Service (SCS), has recommended two alternative methods for estimating time of concentration. The first is based on a relationship for basin lag time, defined as the time between the center of mass of rainfall and peak discharge of the corresponding hydrograph. The agency estimated that time of concentration is typically 5/3 of the lag time. Combining these two relationships yields (SCS, 1986)


    where the curve number, CN, can be obtained from Chapter 3. For a highly urban area, the time of concentration can be multiplied by an adjustment factor, M, determined by


    where p is the percent imperviousness or percent of main channels that are hydraulically improved beyond natural conditions. The NRCS recommends that Equation 5-15 be limited to homogeneous watersheds that have curve numbers between 50 and 95 and that are smaller than 2,000 acres (800 ha).

    The second method recommended by the NRCS is the upland method, which can be expressed as


    where V is the velocity of overland flow in fps or m/s, which can be estimated from Figure 5-1 based on land use and surface slope in percent. For cases where the primary flow path can be divided into various segments having different slopes or land uses, tc should be evaluated by


    ( )=


    1jjjc VL60


    SKV =

    (5-18) where j refers to a particular flow segment.

    Figure 5-1: Velocities for use in the upland method (SCS, 1986)

    An alternative to using Figure 5-1 is to express velocity as


    Velocity (fps)


    e (%





    c SL0078.0t =

    where V is in fps; K is conveyance, listed in Table 5-2 for various surfaces; and S is in ft/ft. The basis for these values, as well as the basis for Figure 5-1, are various assumed combinations of flow geometry and surface roughness.

    Table 5-2: Conveyance values for overland flow

    Surface description K

    Forestland 0.7 2.5

    Grass 1.0 2.1

    Short-grass prairie 7.0

    Natural rangeland 1.3

    Paved area 20.4 Source: Adapted from McCuen (1998)

    Care should be taken when applying Equation 5-19 for slopes that exceed approximately four percent. In these instances, velocity profiles become more complex and V tends to be overestimated. Kirpich Equation

    The Kirpich equation for time of concentration can be expressed as (Kirpich, 1940)


    This relationship was originally developed from SCS data for well-defined and relatively steep channels draining small- to moderate-sized watersheds (i.e., < 100 acres), but it often yields satisfactory results for overland flow on bare soils and for areas up to 200 acres (80 ha). Note that for more general cases of overland flow, Rossmiller (1980) recommends that tc be multiplied by an adjustment factor 2.0. For concrete or asphalt surfaces, the adjustment factor reduces to 0.4. Izzard Equation

    Based on a series of laboratory experiments by the Bureau of Public Roads, Izzard (1946) proposed the following relationship for time of concentration for roadway and turf surfaces:


    ( )3231


    c iSLci0007.0025.41t +=

    ( )31


    c SLC1.139.0t =

    ( )S


    c =


    where c is a retardance factor that ranges from 0.007 for smooth pavement to 0.012 for concrete and to 0.06 for dense turf. The method is designed for applications in which the product of intensity (in/hr) and flow length (ft) is less than 500. In addition, application of 5-21 requires an iterative solution, similar to that of the kinematic wave model, since i is dependent on time of concentration. Kerby Equation

    Kerby (1959) defined flow length as the straight-line distance from the most distant point of a basin to its outlet, measured parallel to the surface slope. Based on this definition, time of concentration can be evaluated as


    This relationship is not commonly used and has the most limitations. It was developed based on watersheds less than 10 acres (4 ha) in size and having slopes less than one percent. It is generally applicable for flow lengths less than 1,000 ft (300 m). FAA Method

    The Federal Aviation Administration (FAA, 1970) used airfield drainage data assembled by the U.S. Army Corps of Engineers to develop an estimate for time of concentration. The method has been widely used for overland flow in urban areas and can be expressed as


    where C is a dimensionless runoff coefficient, which can be obtained from Table 5-3.


    Table 5-3: Runoff coefficients for 2 to 10 year return periods

    Description of drainage area Runoff coefficient


    Downtown Neighborhood

    0.70 - 0.95 0.50 - 0.70

    Residential Single-family Multi-unit detached Multi-unit attached

    0.30 - 0.50 0.40 - 0.60 0.60 - 0.75

    Suburban 0.25 - 0.40

    Apartment dwelling 0.50 - 0.70

    Industrial Light Heavy

    0.50 - 0.80 0.60 - 0.90

    Parks and cemeteries 0.10 - 0.25

    Railroad yards 0.20 - 0.35

    Unimproved areas 0.10 - 0.30

    Pavement Asphalt Concrete Brick

    0.70 - 0.95 0.80 - 0.95 0.75 - 0.85

    Roofs 0.75 - 0.95

    Lawns Sandy soils

    Flat (2%) Average (2 7 %) Steep ( 7%)

    Heavy soils Flat (2%) Average (2 7 %) Steep ( 7%)

    0.05 - 0.10 0.10 - 0.15 0.15 - 0.20

    0.13 - 0.17 0.18 - 0.22 0.25 - 0.35

    Source: Adapted from ASCE (1992) Yen and Chow Method

    Yen and Chow (1983) proposed the following expression for evaluation of time of concentration:



    21Yc SNLKt



    where KY ranges from 1.5 for light rain (i < 0.8 in/hr) to 1.1 for moderate rain (0.8 < i < 1.2 in/hr), and to 0.7 for heavy rain (i > 1.2 in/hr); and N is an overland texture factor,...


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